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Rate of Convergence of Positive Series

Authors:

O B Skaskiv

Institution:

(1) Lviv National University, Lviv, Ukraine

Abstract:

We investigate the rate of convergence of series of the form

$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } a_n e^{x\lambda _n + \tau (x)\beta _n } ,\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1$

where λ = (λ_n), 0 = λ₀ < λ_n ↑ + ∞, n → + ∞, β = {β_n: n ≥ 0} ⊂ ℝ₊, and τ(x) is a nonnegative function nondecreasing on 0; +∞), and

$F(x) = \mathop \sum \limits_{n = 0}^{ + \infty } a_n f(x\lambda _n ),\quad a_n \geqslant 0,\quad n \geqslant 1,\quad a_0 = 1,$

where the sequence λ = (λ_n) is the same as above and f (x) is a function decreasing on 0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on 0; +∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.

Keywords:

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